Radical Sylvester-Gallai Theorem for Cubics

Let F = tF 1 , . . . , F m u be a finite set of irreducible homogeneous multivariate polynomials of degree at most 3 such that F i does not divide F j for i ‰ j. We say that F is a cubic radical Sylvester-Gallai configuration if for any two distinct F i , F j there exists a third polynomial F k such that whenever F i , F j vanish, F k also vanishes. In particular, for any two indices i, j P [m], there exists k P [m]zti, ju such that F k P rad(F i , F j ).

We prove that any cubic radical Sylvester-Gallai configuration is low-dimensional, that is

This solves a conjecture of Gupta [Gup14] in degree 3 and generalizes the result in [Shp20], which proved that quadratic radical Sylvester-Gallai configurations are low-dimensional. Our result takes us one step closer towards solving the non-linear Sylvester-Gallai conjectures of Gupta [Gup14], which would yield the first deterministic polynomial time algorithm for the PIT problem for depth-4 circuits of bounded top and bottom fanins.

To prove our Sylvester-Gallai theorem, we develop several new tools combining techniques from algebraic geometry and elimination theory. Among our technical contributions, we prove a structure theorem characterizing non-radical ideals generated by two cubic forms, generalizing the structure theorems of [HP94, CTSSD87, Shp20]. Moreover, building upon the groundbreaking work [AH20a], we introduce the notion of wide Ananyan-Hochster algebras and show that these algebras allow us to transfer the local conditions of Sylvester-Gallai configurations into global conditions.